This lack of mathematically rigorous well-posedness results for the afore-mentioned PDEs is not merely of academic interest but impacts the practical computation of both forward and Bayesian inverse problems as in practice, one approximates the posterior measure by sampling it using Markov Chain Monte Carlo (MCMC) algorithms of the Metropolis-Hastings type or their variants [1]}. This, in turn, entails evaluating the forward map \(\mathcal {L}\)  (REF ) multiple times. However, in general, one has to numerically simulate this forward map i.e., replace it with an approximation \(\mathcal {L}^{\Delta } \approx \mathcal {L}\) , with \(\Delta \)  being a numerical parameter, such as the mesh size. Related to the lack of well-posedness of the underlying solution operator, it has been shown in recent papers such as [2]}, [3]}, [4]}, [5]} that standard numerical approximations \(\mathcal {L}^{\Delta }\)  may not necessarily converge on mesh refinement (as \(\Delta \rightarrow 0\) ) or converge too slowly to be of any practical interest. Consequently, even the rationale for numerical approximation of Bayesian inverse problems for these ill-posed PDEs is completely unclear.
